Well, to start with, there isn't much in the way of trees at
Lookout Cay. There is a pretty beach, but you can get that at many of the existing ports of call. The appeal of a "private island" is that there are no headaches with transportation, locals or food. Lookout Cay fails in two of the three.
Transportation is a big headache, with the long, hot pier with no shade or seating, and followed by the tram. It's easier and takes no longer to cross a typical port of call island via taxi to get to a great beach, then it does to get from the ship at Lookout Cay to its beach. And at normal beaches, there is more access to loungers with a view of the water.
The food at Lookout Cay has serious sanitary issues, given the fly infestation. A private island is supposed to alleviate concerns about unsanitary food. Not here. Food is also unavailable later in the afternoon, despite the fact that this island tends to feature a late morning arrival and an early evening departure. They still close down all of the food and smoothie stations around the same time as on Castaway, despite the fact that Castaway operates on an earlier schedule.
Lookout Cay is probably safe from crime and other nuisances public islands can be prone to. This is the one real advantage. However, it isn't much of an advantage, because many Caribbean islands are safe and don't feature tourists being harassed by vendors.